Advanced Microeconomics

Transcription

1 Simple economy with production November 27, 2011

2 Introducing production We start with a simplest possible setting: single consumer and a single rm. Two goods: labor/leisure and a consumption good produced by the rm This is often referred to as Robinson Crusoe economy.

3 The rm The rm uses labor to produce the consumption goods. Increasing and strictly concave production function f (z), where z is the labour input. To produce output the rm purchases some labour from the consumer. The rm takes market prices as given.

4 The rm The rm solves the standard prot maximization problem: Max pf (z) wz. z 0 Given prices (p, w) the rms optimal: labor demand is z(p, w) [labor demand function] output is q(p, w) [supply function] prots are π(p, w). [prot function]

5 The rm

6 Consumer Consumer is the sole owner of the rm and receives all the prots. Consumer (labour owner) is hired by the rm at the same time. He has endowment L of labour. Consumer preferences are represented by a utility function u(x 1, x 2 ). x 1 is the consumption of leisure, therefore labour supply is L x 1 x 2 is the consumption good The consumer problem given prices is: Max u(x 1, x 2 ) s.t. px 2 w( L x1 ) + π(p, w). (x1,x2) R 2 + Budget constraint reects the two sources of income for a consumer.

7 Consumer

8 Demand and equilibrium Consumer optimal demands at prices (p, w) are denoted by: (x 1 (p, w), x 2 (p, w)) A Walrasian equilibrium in this economy involves a price vector (p, w ) at which the consumption and labour markets clear: x 2 (p, w ) = q(p, w ) z(p, w ) = L x1 (p, w ).

9 The rst order conditions for an interior optimum From the consumer maximization problem: w p = MU 1 MU 2 From the rms prot maximization: pmp z = w w p = MP z Therefore: MP z = MU 1 MU 2 (remember that consumption of good 1 is L z).

10 Equilibrium

11 Welfare properties A particular consumption-leisure combination can arise in a competitive equilibrium if and only if it maximizes the consumers' utility subject to the economy's technological and endowment constraints. The same allocation as the social planner solution (under certain conditions, such as existence of prot maximizers). 1st and 2nd welfare theorem hold. For 2nd we need convexity of preferences and strict convexity of the aggregate production set.

12 The social planner solution We can nd the optimal use of resources in our economy by maximizing the utility function of the consumer: Max u(x 1, x 2 ) x1,x2 subject to the resource constraint L = x1 + z x 1 = L z subject to technology constraint:x 2 = q = f (z) Subsitute to the utility function to get a function of z solely: Max u( L z, f (z)). z So the social planner will make the agent work the optimal amount of hours.

13 The social planner solution First order condition u( L z, f (z)) z = u(x 1, x 2 ) x 1 f (z) + z u(x 1, x 2 ) x 2 = 0 MU 1 + MP z MU 2 = 0 MP z = MU 1 MU 2 So as long as the interior solution to both problems (comp. eq. and soc. plan.) exist, they are the same.

14 Three examples with dierent technology Example 1 (constant returns to scale): Suppose we have one consumer with a Cobb-Douglas utility function for consumption x 2 and leisure x 1 : u(x 1, x 2 ) = a ln x 1 + (1 a) ln x 2. Endowment of leisure is one L = 1. The rm has with a constant-returns-to-scale technology q = f (z) = z. Find the competitive equilibrium. Find the Pareto optimal allocation through a social planner problem. Do the two coincide?

15 DRS Example 2 (decreasing returns to scale): Suppose we have one consumer with a Cobb-Douglas utility function for consumption x 2 and leisure x 1 : u(x 1, x 2 ) = a ln x 1 + (1 a) ln x 2. Endowment of leisure is L = 1. Consumer can supply its leisure to the rm with a decreasing-returns-to-scale technology q = z. Find the competitive equilibrium. Find the Pareto optimal allocation through a social planner problem. Do the two coincide?

16 IRS Example 3 (increasing returns to scale): Suppose we have one consumer with a Cobb-Douglas utility function for consumption x 2 and leisure x 1 : u(x 1, x 2 ) = a ln x 1 + (1 a) ln x 2. Endowment of leisure is L = 1. Consumer can supply its leisure to the rm with an increasing-returns-to-scale technology q = L 2. Does the competitive equilibrium exist? Find the Pareto optimal allocation through a social planner problem. Do the two coincide?

17 Welfare properties